Models for Hybrid Systems Bart De Schutter TU Delft, The - - PDF document

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IEEE CSS Technical Committee on Hybrid Systems

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www.ist-hycon.org www.unisi.it

1 HYCON PhD School on Hybrid Systems

st

Siena, July 1 9-22, 2005 - Rectorate of the University of Siena

Models for Hybrid Systems Bart De Schutter

TU Delft, The Netherlands

b.deschutter@its.tudelft.nl

1st HYCON Summer School on Hybrid Systems Siena, Italy, July 13–18, 2005

Models for Hybrid Systems

Bart De Schutter (b.deschutter@dcsc.tudelft.nl) Delft Center for Systems and Control Delft University of Technology

Models for Hybrid Systems

Overview

• 1. Hybrid systems — Definition, examples & challenges
• 2. Hybrid system models — Overview & issues
• 3. Models for event-driven systems — Automata
• 4. Hybrid automata
• 5. PWA systems and related model classes (MLD, LC, MMPS)
• 6. Timed automata
• 7. (Timed Petri nets)
• 8. Summary
• 1. Hybrid systems

1.1 Informal definition

• Hybrid = combination of continuous and discrete dynamics
• Temperature control system:
• n mode

˙ T = fon(T,w)

• ff mode

˙ T = foff(T,w) T > Tupp T < Tlow

hs models.1

1.2 More formal definition

• System can be in one of several modes
• In each mode: behavior described by system of

difference or differential equations

• Mode switches due to occurrence of “events”

˙ x1 = f1(x1,u) ˙ x2 = f2(x2,u) ˙ x3 = f3(x3,u) y = g1(x1,u) y = g2(x2,u) y = g3(x3,u)

hs models.2

1.2 More formal definition (continued)

• At switching time instant:

→ possible state reset or state dimension change

• Mode transitions may be caused by

– external control signal – internal control signal – dynamics of system itself (crossing of boundary in state space)

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1.3 Examples

• Hierarchical control in process industry
• Telecommunication systems
• Manufacturing systems
• Air traffic coordination and control
• Batch processes

(e.g. beer brewing)

Human intervention in smooth systems → hybrid

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1.3 Examples (continued)

• Traffic control
• Automatic platooning
• Evolution of rigid bodies

(contact/no contact)

• Electrical networks (switching, diodes)
• Fermentation process

(lag, growth, stationary, inactivation)

• Saturation, hysteresis
• Actuator and sensor failures

Switching between dynamical regimes → hybrid

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1.4 Challenges

• Analysis and control
• Nowadays:

– often still heuristic & ad-hoc – often focus still exclusively on either continuous or discrete dynamics → structured approach necessary

• Consider hybrid nature of systems
• Combination of systems & control, computer science,

mathematics, and simulation

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• 2. Hybrid system models

2.1 Introduction

• Continuous-state / discrete-state
• Continuous-time / discrete-time
• Time-driven / event-driven

– time-driven → state changes as time progresses, i.e., continuously (for CT), or at every tick of clock (for DT) – event-driven → state changes due to occurrence of event event: ∗ start or end of an activity ∗ asynchronous (occurrence times not necessarily equidistant) Combinations → “hybrid”

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2.2 Models for time-driven systems

• Continuous-time time-driven systems:

˙ x(t) = f(x(t),u(t)) y(t) = g(x(t),u(t))

• Discrete-time (or sampled) time-driven systems:

x(k +1) = f(x(k),u(k)) y(k) = g(x(k),u(k)

hs models.8

2.3 Models for event-driven systems

• Z

Automata

• Z

Petri nets

• (max,+) algebraic models
• Markov chains / Markov processes
• Extended state machines
• Generalized semi-Markov processes
• Networks of waiting queues
• . . .

⇒ no general framework Note: see also lecture on “Discrete-event modeling and diagnosis of quantized systems”

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2.3 Models for event-driven systems (continued)

• No general framework (similar situation for hybrid systems)
• Basic trade-off:

modeling power ↔ decision power ⇒ application-specific

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2.4 Models for hybrid systems

• Z

Timed or hybrid Petri nets

• Differential automata
• Z

Hybrid automata

• Brockett’s model
• Z

Mixed logical dynamic models

• Duration calculus
• Real-time temporal logics
• Timed communicating sequential processes
• Switched bond graphs
• ../..

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2.4 Models for hybrid systems (continued)

• Computer simulation models
• Predicate calculus
• Z

Piecewise-affine models

• Z

Timed automata

• . . .

Note: focus in this lecture is on non-stochastic models (see also lectures on “Stochastic Hybrid Systems”)

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2.5 Models for hybrid systems — Issues ⇒ no general modeling & analysis framework modeling power ↔ decision power + computational complexity (NP-hard, undecidable) ⇒ special subclasses hierarchical / modular approach

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Intermezzo: Undecidable and NP-hard problems

• Undecidable problems

→ no algorithm at all can be given for solving the problem in general

• NP-complete and NP-hard problems

– decision problem: solution is either “yes” or “no” e.g., traveling salesman decision problem: Given a network of cities, intercity distances, and a number B, does there exist a tour with length B? – search problem e.g., traveling salesman problem: Given a network of cities, intercity distances, what is the shortest tour?

intermezzo.1

P and NP-complete decision problems

• time complexity function T(n): largest amount of time needed to

solve problem instance of size n (worst case!)

• polynomial time algorithm:

T(n) |p(n)| for some polynomial p → class P: solvable by polynomial time algorithm

• nondeterministic computer:

– guessing stage (tour) – checking stage (compute length of tour + compare it with B) → class NP: “nondeterministically polynomial ” i.e., time complexity of checking stage is polynomial

intermezzo.2

P and NP-complete decision problems

• Each problem in NP can be solved in exponential time: T(n) 2nk
• NP-complete problems: “hardest” class in NP:

– any NP-complete problem solvable in polynomial time ⇒ every problem in NP solvable in polynomial time – any problem in NP intractable ⇒ NP-complete problems also intractable NP P NP-complete if P=NP

intermezzo.3

NP-hard problems

• decision problem is NP-complete ⇒ search problem is NP-hard
• NP-hard problems: at least as hard as NP-complete problems

– NP-complete (decision problem) → solvable in polynomial time if and only if P = NP – NP-hard (search problem) → cannot be solved in polynomial time unless P = NP

intermezzo.4

Examples of NP-hard and undecidable problems

• Consider simple hybrid system:

x(k +1) =

• A1x(k)

if cTx(k) 0 A2x(k) if cTx(k) < 0 → deciding whether system is stable or not is NP-hard

• Given two Petri nets, do they have the same reachability set?

→ undecidable

intermezzo.5

Back to the main topic — Hybrid system models

• Many modeling frameworks for hybrid systems

⇒ trade-off: modeling power ↔ decision power, tractability

• Hybrid automata:

– very general, high modeling power, but low decision power – analysis and control → computationally hard (NP-hard, undecidable problems)

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Hybrid system models (continued)

• Computer simulation and verification tools: Modelica, HyTech,

KRONOS, Chi, 20-sim, UPPAAL, . . . + simulation models can represent plant with high degree of detail (high modeling power)

• computationally very demanding for large systems
• difficult to understand from simulation how behavior depends
• n model parameters
• In this lecture: special classes of hybrid systems for which

tractable analysis and control design techniques are available (→ see next lectures)

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• 3. Models for event-driven systems

3.1 Automata Automaton Automaton is defined by triple Σ = (Q,U ,φ) with

• Q: finite or countable set of discrete states
• U : finite or countable set of discrete inputs (“input alphabet”)
• φ : Q ×U → P(Q): partial transition function.

where P(Q) is power set of Q (set of all subsets) Finite automaton: Q and U finite

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3.1 Automata (continued) Evolution of automaton

• Given state q ∈ Q and discrete input symbol u ∈ U ,

transition function φ defines collection of next possible states: φ(q,u) ⊆ Q

• If each set of next states has 0 or 1 element:

→ “deterministic” automaton

• If some set of next states has more than 1 element:

→ “non-deterministic” automaton

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3.1 Automata (continued) Deterministic automaton qidle qbusy qdown α β γ δ φ(qbusy,β) = {qidle} φ(qidle,α) = {qbusy} φ(qbusy,γ) = {qdown} φ(qdown,δ) = {qidle}

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3.1 Automata (continued) Non-deterministic automaton q1 q2 α α β φ(q1,α) = {q1,q2} φ(q2,β) = {q1}

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• 4. Hybrid automata

4.1 Definition Hybrid automaton H is collection H = (Q,X, f,Init,Inv,E,G,R) with

• Q = {q1,...,qN} is finite set of discrete states or modes
• X = Rn is set of continuous states
• f : Q×X → X is vector field
• Init ⊆ Q×X is set of initial states
• Inv : Q → P(X) describes the invariants
• E ⊆ Q×Q is set of edges or transitions
• G : E → P(X) is guard condition
• R : E → P(X ×X) is reset map

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Hybrid automaton H = (Q,X, f,Init,Inv,E,G,R)

• Hybrid state: (q,x)
• Evolution of continuous state in mode q: ˙

x = f(q,x)

• Invariant Inv: describes conditions that continuous state has to

satisfy at given mode

• Guard G: specifies subset of state space where certain transition

is enabled

• Reset map R: specifies how new continuous states are related to

previous continuous states

hs models.21

(q0,x0) ∈ Init q0 ˙ x = f(q0,x) x ∈ Inv(q0) q1 ˙ x = f(q1,x) x ∈ Inv(q1) q2 ˙ x = f(q2,x) x ∈ Inv(q2) G(q0,q1) G(q1,q0) G(q1,q2) G(q2,q1) G(q0,q2) G(q2,q0) R(q0,q1) R(q1,q0) R(q1,q2) R(q2,q1) R(q0,q2) R(q2,q0)

hs models.22

Evolution of hybrid automaton

• Initial hybrid state (q0,x0) ∈ Init
• Continuous state x evolves according to

˙ x = f(q0,x) with x(0) = x0 discrete state q remains constant: q(t) = q0

• Continuous evolution can go on as long as x ∈ Inv(q0)
• If at some point state x reaches guard G(q0,q1), then

– transition q0 → q1 is enabled – discrete state may change to q1, continuous state then jumps from current value x− to new value x+ with (x−,x+) ∈ R(q0,q1)

• Next, continuous evolution resumes and whole process is re-

peated

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4.2 Examples of hybrid automata

• 1. Hysteresis
• 2. Water-level monitor

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Hysteresis Control system with hysteresis element in the feedback loop : ˙ x = H(x)+u H Δ −Δ x 1 −1

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Hysteresis (continued) ˙ x = H(x)+u H Δ −Δ x 1 −1 Guard: x Δ Guard: x −Δ H = 1 ˙ x = 1+u x ∈ {x | x Δ} H = −1 ˙ x = −1+u x ∈ {x | x −Δ}

hs models.26

Water-level monitor

• variables:

– y(t): water level, continuous – x(t): time elapsed since last signal was sent by monitor, cont. – P(t): status of pump, ∈ {on,off} – S(t): nature of signal last sent by monitor, ∈ {on,off}

• dynamics of system:

– water level rises 1 unit per second when pump is on and falls 2 units per second when pump is off – when water level rises to 10 units, monitor sends switch-off signal; after delay of 2 seconds pump turns off – when water level falls to 5 units, monitor sends switch-on sig- nal; after delay of 2 seconds pump switches on

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x = 2 x = 2 x := 0 x := 0 y = 5 y = 10

mode: on,on

˙ x = 1 ˙ y = 1 y 10

mode: on,off

˙ x = 1 ˙ y = 1 x 2

mode: off,off

˙ x = 1 ˙ y = −2 y 5

mode: off,on

˙ x = 1 ˙ y = −2 x 2 y: water level x: time since last signal

hs models.28

• 5. PWA systems and related model classes
• 1. Piecewise affine systems (PWA)
• 2. Mixed Logical Dynamical systems (MLD)
• 3. Linear Complementarity systems (LC)
• 4. Max-Min-Plus-Scaling systems (MMPS)
• 5. Equivalence of MLD, LC, ELC, PWA and MMPS systems

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5.1 Piecewise affine (PWA) systems

• PWA systems are described by

x(k +1) = Aix(k)+Biu(k)+ fi y(k) = Cix(k)+Diu(k)+gi for x(k) u(k)

• ∈ Ωi, i = 1,...,N
• Ω1,...,ΩN: convex polyhedra (i.e., given by finite number of linear

inequalities) in input/state space, non-overlapping interiors

• PWA can be used as approximation of nonlinear model

x(k +1) = Nx(x(k),u(k)) y(k) = Ny(x(k),u(k)) → “simplest” extension of linear systems that can still model non-linear & non-smooth processes with arbitrary accuracy + are capable of handling hybrid phenomena

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Example of PWA model Integrator with upper saturation: x(k +1) =

• x(k)+u(k)

if x(k)+u(k) 1 1 if x(k)+u(k) 1 y(k) = x(k) u(k) x(k) x(k)+u(k) < 1 x(k)+u(k) > 1

hs models.31

5.2 Mixed Logical Dynamical (MLD) systems Preliminaries

• Boolean operators:

∧ (and), ∨ (or), ∼ (not), ⇒ (implies), ⇔ (iff), ⊕ (xor) X1 X2 X1 ∧X2 X1 ∨X2 ∼X1 X1 ⇒ X2 X1 ⇔ X2 X1 ⊕X2 T T T T F T T F T F F T F F F T F T F T T T F T F F F F T T T F

• Properties:

– X1 ⇒ X2 is same as ∼X1 ∨X2 – X1 ⇒ X2 is same as ∼X2 ⇒ ∼X1 – X1 ⇔ X2 is same as (X1 ⇒ X2)∧(X2 ⇒ X1)

hs models.32

• Associate with literal Xi logical variable δi ∈ {0,1}:

δi = 1 iff Xi = T, δi = 0 iff Xi = F → compound statement can be transformed into linear integer program

• Examples:

* X1 ∧X2 equivalent to δ1 = δ2 = 1 * X1 ∨X2 equivalent to δ1 +δ2 1 * ∼X1 equivalent to δ1 = 0 * X1 ⇒ X2 equivalent to δ1 −δ2 0 * X1 ⇔ X2 equivalent to δ1 −δ2 = 0 * X1 ⊕X2 equivalent to δ1 +δ2 = 1

• For f : Rn → R and x ∈ X with X bounded, define

M

def

= max

x∈X f(x)

m

def

= min

x∈X f(x) hs models.33

• Equivalences:

* [f(x) 0]∧[δ = 1] true iff f(x)−δ −1+m(1−δ) * [f(x) 0]∨[δ = 1] true iff f(x) Mδ * ∼[f(x) 0] true iff f(x) ε (with ε machine precision) * [f(x) 0] ⇒ [δ = 1] true iff f(x) ε +(m−ε)δ * [f(x) 0] ⇔ [δ = 1] true iff

• f(x) M(1−δ)

f(x) ε +(m−ε)δ

• Product δ1δ2 can be replaced by auxiliary variable δ3 = δ1δ2

Since [δ3 = 1] ⇔ [δ1 = 1]∧[δ2 = 1], δ3 = δ1δ2 is equivalent to ⎧ ⎪ ⎨ ⎪  −δ1 +δ3 0 −δ2 +δ3 0 δ1 +δ2 −δ3 1

hs models.34

• Product δ f(x) can be replaced by auxiliary real variable

y = δ f(x) with [δ = 0] ⇒ [y = 0], [δ = 1] ⇒ [y = f(x)],

• r equivalently

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪  y Mδ y mδ y f(x)−m(1−δ) y f(x)−M(1−δ)

hs models.35

Mixed logical dynamical (MLD) systems

• x(k +1) = Ax(k)+B1u(k)+B2δ(k)+B3z(k)

y(k) = Cx(k)+D1u(k)+D2δ(k)+D3z(k) E1x(k)+E2u(k)+E3δ(k)+E4z(k) g5,

• x(k) = [xrT(k) xbT(k)]T with xr(k) real-valued, xb(k) boolean

z(k): real-valued auxiliary variables δ(k): boolean auxiliary variables

• Applications: PWA systems, systems with discrete inputs, quali-

tative inputs, bilinear systems, finite state machines

• Reference: A. Bemporad and M. Morari, “Control of systems integrating

logic, dynamics, and constraints,” Automatica, vol. 35, no. 3, pp. 407–427, March 1999.

hs models.36

Example of an MLD system

• Consider PWA system:

x(k +1) =

• 0.8x(k)+u(k)

if x(k) 0 −0.8x(k)+u(k) if x(k) < 0 where x(k) ∈ [−10,10] and u(k) ∈ [−1,1]

• Associate binary variable δ(k) to condition x(k) 0

such that [δ(k) = 1] ⇔ [x(k) 0] or −mδ(k) x(k)−m −(M +ε)δ −x−ε where M = −m = 10, and ε is machine precision

• PWA system can be rewritten as

x(k +1) = 1.6δ(k)x(k)−0.8x(k)+u(k)

hs models.37

• x(k +1) = 1.6δ(k)x(k)−0.8x(k)+u(k)
• Define new variable z(k) = δ(k)x(k) or

z(k) Mδ(k) z(k) mδ(k) z(k) x(k)−m(1−δ(k)) z(k) x(k)−M(1−δ(k))

• PWA system now becomes

x(k +1) = 1.6z(k)−0.8x(k)+u(k) subject to linear constraints above → MLD

hs models.38

5.3 Linear Complementarity (LC) systems

• LC systems:

x(k +1) = Ax(k)+B1u(k)+B2w(k) y(k) = Cx(k)+D1u(k)+D2w(k) v(k) = E1x(k)+E2u(k)+E3w(k)+g4 0 v(k) ⊥ w(k) 0

• v(k), w(k): “complementarity variables” (real-valued)
• Applications: constrained mechanical systems, electrical networks

with ideal diodes, boost converter, dynamical systems with PWA relations, variable-structure systems, projected dynamical sys- tems

• Example: two-carts system (continuous-time LC system)

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Example of an LC system Two-carts system

• Two carts connected by spring
• Left cart attached to wall by spring;

motion constrained by completely inelastic stop Stop is placed at equilibrium position of left cart

• Masses of carts and spring constants = 1

x1 x2

hs models.40

Two-carts system (continued) x1 x2

• x1, x2: deviations of left and right cart from equilibrium position
• x3,x4: velocities of left and right cart
• z: reaction force exerted by stop
• Evolution: ˙

x1(t) = x3(t) ˙ x2(t) = x4(t) ˙ x3(t) = −2x1(t)+x2(t)+z(t) ˙ x4(t) = x1(t)−x2(t)

hs models.41

Two-carts system (continued) To model stop: x1 x2

• define w(t) = x1(t)
• w(t) 0 (since is position of left cart w.r.t. stop)
• force exerted by stop can act only in positive direction → z(t) 0
• if left cart not at stop (w(t) > 0), reaction force vanishes: z(t) = 0
• if z(t) > 0 then cart must necessarily be at the stop: w(t) = 0

0 w(t)⊥z(t) 0 → (continuous-time) LC system

hs models.42

5.4 Max-Min-Plus-Scaling (MMPS) systems

• Max-min-plus-scaling expression:

f := xi|α|max(fk, fl)|min(fk, fl)|fk + fl|β fk with α, β ∈ R and fk, fl again MMPS expressions.

• Example: 5x1 −3x2 +7+max(min(2x1,−8x2),x2 −3x3)
• MMPS systems:

x(k +1) = Mx(x(k),u(k),d(k)) y(k) = My(x(k),u(k),d(k)) Mc(x(k),u(k),d(k)) c, with Mx, My, Mc MMPS expressions

• d(k): real-valued auxiliary variables

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5.4 Max-Min-Plus-Scaling (MMPS) systems (continued)

• Applications:

– discrete-event systems (also max-plus) – traffic-signal controlled intersection – railway networks – manufacturing systems – systems with soft & hard synchronization constraints – logistic systems

hs models.44

Example of MMPS system

• Integrator with upper saturation:

x(k +1) =

• x(k)+u(k)

if x(k)+u(k) 1 1 if x(k)+u(k) 1 y(k) = x(k) can be recast as x(k +1) = min(x(k)+u(k),1) y(k) = x(k)

hs models.45

5.5 Equivalence of MLD, LC, PWA and MMPS systems Equivalence between model classes A and B: for each model ∈ A there exists model ∈ B with same input/output behavior (+ vice versa) MLD, LC, PWA and MMPS systems are equivalent:

* * * *

MLD LC PWA MMPS

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Equivalence of MLD, LC, PWA and MMPS systems (cont.)

• Each subclass has own advantages:

– stability criteria for PWA – control and verification techniques for MLD – control techniques for MMPS – conditions of existence and uniqueness of solutions for LC → transfer techniques from one class to other

• It depends on the application which class is best suited
• Reference: W.P

.M.H. Heemels, B. De Schutter, and A. Bemporad, “Equiva- lence of hybrid dynamical models,” Automatica, vol. 37, no. 7, pp. 1085–1091, July 2001.

hs models.47

• 6. Timed automata
• Timed automata involve simple continuous dynamics:

– all differential equations of form ˙ x = 1, – all invariants, guards, etc. involve comparison of real-valued states with constants (e.g., x = 1, x < 2, x 0, etc.)

• Timed automata are limited for modeling physical systems.
• However, very well suited for encoding timing constraints such as

“event A must take place at least 2 seconds after event B and not more than 5 seconds before event C”

• Applications: multimedia, Internet, audio protocol verification

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6.1 Rectangular sets

• Subset of Rn set is called rectangular if can be written as finite

boolean combination of constraints of form xi a, xi < b, xi = c, xi d, xi > e

• Rectangular sets are “rectangles” or “boxes” in Rn whose sides

are aligned with the axes, or unions of such rectangles/boxes

• Examples:

– {(x1,x2) | (x1 0)∧(x1 2)∧(x2 1)∧(x2 2)} – {(x1,x2) | ((x1 0)∧(x2 = 0))∨((x1 = 0)∧(x2 0))} – empty set (e.g., ∅ = {(x1,x2) | (x1 > 1)∧(x1 0))}

• However, set {(x1,x2) | x1 = 2x2} is not rectangular

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6.2 Timed automaton

• Timed automaton is hybrid automaton with following characteris-

tics: – automaton involves differential equations of form ˙ xi = 1; continuous variables governed by this differential equation are called “clocks” or “timers” – sets involved in definition of initial states, guards, and invari- ants are rectangular sets – reset maps involve either rectangular set, or may leave certain states unchanged

hs models.50

6.3 Example of timed automaton q1 ˙ x1 = 1 ˙ x2 = 1 x2 3 q2 ˙ x1 = 1 ˙ x2 = 1 x1 5 x2 > 2 x1 > 4 x1 := 3∧x2 := 0 x1 := 0 x1 = x2 = 0

hs models.51

• 7. Timed Petri nets

7.1 Petri nets

• Graphical representation: bipartite directed multigraph

– places (circles) → activities – transitions (bars) → events, actions p1 p2 p3 p4 p5 t1 t2 t3 t4 t5

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• marking → tokens are assigned to places
• execution of Petri net:

– transition enabled if all input places (•t) contain at least 1 token – enabled transition can fire: ∗ one token is removed from each input place (•t) ∗ one token is deposited in each output place (t•) p1 p1 p2 p2 p3 p3 p4 p4 p5 p5 t1 t1 t2 t2 t3 t3 t4 t4 t5 t5

• synchronization & choice

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7.2 Timed Petri nets

• Untimed Petri net describes order in which events can occur,

but no timing

• Timed Petri → timing, transition should be executed within cer-

tain time interval after it becomes enabled – discrete state variables (markings, mθ(p)) – continuous state variables (arrival times, Mθ(p))

• Mθ(p) := {θ1,...,θmθ(p)} with arrival times θ1 θ2 ... θmθ(p) of

mθ(p) tokens in place p

• For each transition t we define interval [L(t),U(t)]

hs models.54

7.2 Timed Petri nets (continued)

• Transition t becomes enabled at

max

p∈•t minMθ(p)

• Then transition t may fire at some time

θ ∈ [max

p∈•t minMθ(p)+L(t),max p∈•t minMθ(p)+U(t)]

provided t is enabled during whole interval

• If enabling condition is still valid at final time of firing interval, then

transition is forced to fire

• Many techniques for untimed Petri nets can be extended to timed

Petri nets

• However, many problems are undecidable or NP-hard

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• 8. Summary
• Hybrid: combination of discrete-event and continuous dynamics
• Many modeling frameworks

– trade-off: modeling power vs. decision power – application specific

• In the next lectures: properties, analysis, control, identification,

fault diagnosis, and applications

hs models.56

Selected references

• P

.J. Antsaklis and A. Nerode, eds., “Special issue on hybrid systems,” IEEE Transactions on Automatic Control, vol. 43, no. 4, Apr. 1998.

• A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics,

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